Let $k$ be a field. There are several ways to make $k$ into a $k[X]$-module; for example, by fixing $\alpha \in k$ and taking $P(X) \cdot t := P(\alpha)t$ for all $t \in k$. Are there other obvious ways to make $k$ into a $k[X]$-module? Can we actually give a nice descriptions of all possible such structures?
What can be said about the $k[X_1,X_2,\cdots,X_n]$-module structures on $k$ for $n \geq 2$?
All interesting $k[x]$-module structures on $k$ can be given by a pair $(\phi,\alpha)$ where $\phi$ is an automorphism of $k$ and $\alpha\in k$. Then the module action is determined by:
$$P(x)\cdot t = \phi(P(\phi^{-1}(\alpha)))t$$
If you want $k[x]$ to act on $k$ consistent with the "obvious" action of $k$ on $k$, then $\phi = id_k$ and arbitrary $\alpha$ give you all the possible solutions.
This doesn't give all module structures. Let $k_0$ be the least subfield of $k$ containing $0$ and $1$. So $k_0\cong\mathbb Q$ or $k_0\cong\mathbb Z_p$. Let $T$ be an invertible linear operator on the field $k$ when viewed as a vector space over $k_0$. Then you can get an even more general operator given $(\phi,\alpha,T)$:
$$P(x)\cdot t = T^{-1} \phi(P(\phi^{-1}(\alpha)))(Tt)$$
